library(tidyverse) # for data wrangling etc
library(rstanarm) # for fitting models in STAN
library(cmdstanr) # for cmdstan
library(brms) # for fitting models in STAN
library(coda) # for diagnostics
library(bayesplot) # for diagnostics
library(ggmcmc) # for MCMC diagnostics
library(DHARMa) # for residual diagnostics
library(rstan) # for interfacing with STAN
library(emmeans) # for marginal means etc
library(broom) # for tidying outputs
library(tidybayes) # for more tidying outputs
library(ggeffects) # for partial plots
library(broom.mixed) # for summarising models
library(ggeffects) # for partial effects plots
library(bayestestR) # for ROPE
library(see) # for some plots
library(easystats) # for the easystats ecosystem
library(ggridges) # for ridge plots
library(patchwork) # for multiple plots
library(modelsummary) # for data and model summaries
theme_set(theme_grey()) # put the default ggplot theme back
source("helperFunctions.R")Bayesian GLM Part5
1 Preparations
Load the necessary libraries
2 Scenario
Here is a modified example from Quinn and Keough (2002). Day and Quinn (1989) described an experiment that examined how rock surface type affected the recruitment of barnacles to a rocky shore. The experiment had a single factor, surface type, with 4 treatments or levels: algal species 1 (ALG1), algal species 2 (ALG2), naturally bare surfaces (NB) and artificially scraped bare surfaces (S). There were 5 replicate plots for each surface type and the response (dependent) variable was the number of newly recruited barnacles on each plot after 4 weeks.
| TREAT | BARNACLE |
|---|---|
| ALG1 | 27 |
| .. | .. |
| ALG2 | 24 |
| .. | .. |
| NB | 9 |
| .. | .. |
| S | 12 |
| .. | .. |
| TREAT | Categorical listing of surface types. ALG1 = algal species 1, ALG2 = algal species 2, NB = naturally bare surface, S = scraped bare surface. |
| BARNACLE | The number of newly recruited barnacles on each plot after 4 weeks. |
3 Read in the data
Rows: 20 Columns: 2
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (1): TREAT
dbl (1): BARNACLE
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.spc_tbl_ [20 × 2] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
$ TREAT : chr [1:20] "ALG1" "ALG1" "ALG1" "ALG1" ...
$ BARNACLE: num [1:20] 27 19 18 23 25 24 33 27 26 32 ...
- attr(*, "spec")=
.. cols(
.. TREAT = col_character(),
.. BARNACLE = col_double()
.. )
- attr(*, "problems")=<externalptr> day (20 rows and 2 variables, 2 shown)
ID | Name | Type | Missings | Values | N
---+----------+-----------+----------+---------+----------
1 | TREAT | character | 0 (0.0%) | ALG1 | 5 (25.0%)
| | | | ALG2 | 5 (25.0%)
| | | | NB | 5 (25.0%)
| | | | S | 5 (25.0%)
---+----------+-----------+----------+---------+----------
2 | BARNACLE | numeric | 0 (0.0%) | [8, 33] | 20
----------------------------------------------------------| Unique | Missing Pct. | Mean | SD | Min | Median | Max | Histogram | |
|---|---|---|---|---|---|---|---|---|
| BARNACLE | 19 | 0 | 19.8 | 7.4 | 8.0 | 19.5 | 33.0 |  |
| TREAT | N | % | ||||||
| ALG1 | 5 | 25.0 | ||||||
| ALG2 | 5 | 25.0 | ||||||
| NB | 5 | 25.0 | ||||||
| S | 5 | 25.0 |
| TREAT | Unique | Missing Pct. | Mean | SD | Min | Median | Max | Histogram | |
|---|---|---|---|---|---|---|---|---|---|
| BARNACLE | ALG1 | 5 | 0 | 22.4 | 3.8 | 18.0 | 23.0 | 27.0 |  |
| ALG2 | 5 | 0 | 28.4 | 3.9 | 24.0 | 27.0 | 33.0 |  |
|
| NB | 5 | 0 | 15.0 | 4.8 | 9.0 | 14.0 | 22.0 |  |
|
| S | 5 | 0 | 13.2 | 4.5 | 8.0 | 12.0 | 20.0 |  |
|
| TREAT | N | % | |||||||
| ALG1 | 5 | 25.0 | |||||||
| ALG2 | 5 | 25.0 | |||||||
| NB | 5 | 25.0 | |||||||
| S | 5 | 25.0 |
Start by declaring the categorical variables as factor.
Model formula: \[ \begin{align} y_i &\sim{} \mathcal{Pois}(\lambda_i)\\ ln(\mu_i) &= \boldsymbol{\beta} \bf{X_i}\\ \beta_0 &\sim{} \mathcal{N}(3.1, 1)\\ \beta_{1,2,3} &\sim{} \mathcal{N}(0,1)\\ \end{align} \]
where \(\boldsymbol{\beta}\) is a vector of effects parameters and \(\bf{X}\) is a model matrix representing the intercept and treatment contrasts for the effects of Treatment on barnacle recruitment.
4 Exploratory data analysis
The exploratory data analyses that we performed in the frequentist instalment of this example are equally valid here. That is, boxplots and/or violin plots for each population (substrate type).
Conclusions:
- although exploratory data analysis suggests that we might well be fine modelling these data against a Gaussian distribution, a Poisson distribution would clearly be a more natural choice and it would also prevent any non positive predictions.
5 Fit the model
In rstanarm, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.
Priors for model 'day.rstanarm'
------
Intercept (after predictors centered)
~ normal(location = 0, scale = 2.5)
Coefficients
Specified prior:
~ normal(location = [0,0,0], scale = [2.5,2.5,2.5])
Adjusted prior:
~ normal(location = [0,0,0], scale = [5.63,5.63,5.63])
------
See help('prior_summary.stanreg') for more detailsThis tells us:
for the intercept, when the family is Poisson, it is using a normal prior with a mean of 0 and a standard deviation of 2.5. The 2.5 is used for all intercepts. It is often scaled, but only if it is larger than 2.5 is the scaled version kept.
for the coefficients (in this case, just the slope), the default prior is a normal prior centred around 0 with a standard deviation of 2.5. This is then adjusted for the scale of the data by dividing the 2.5 by the standard deviation of the numerical dummy variables for the predictor (then rounded).
- there is no auxiliary prior as we are employing a Poisson distribution.
One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging predictions would ensure that the priors are unlikely to influence the actual predictions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of parameter space that are not supported by the actual data, the sampler can become unstable and have difficulty.
We can draw from the prior predictive distribution instead of conditioning on the response, by updating the model and indicating prior_PD=TRUE. After refitting the model in this way, we can plot the predictions to gain insights into the range of predictions supported by the priors alone.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Conclusions:
- we see that the range of predictions is fairly wide and the predicted means could range from 0 to very large (perhaps too large).
The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]
When defining our own priors, we typically do not want them to be scaled.
If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucked out of thin air):
- \(\beta_0\): normal centred at 3 with a standard deviation of 10
- mean of 3: since
mean(log(day$BARNACLE)) - sd of 1: since
sd(log(day$BARNACLE))
- mean of 3: since
- \(\beta_1\): normal centred at 0 with a standard deviation of 6
- sd of 1: since:
sd(log(day$BARNACLE))/apply(model.matrix(~TREAT, data = day), 2, sd)
- sd of 1: since:
I will also overlay the raw data for comparison.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Now lets refit, conditioning on the data.
In brms, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.
Unlike rstanarm, brms models must be compiled before they start sampling. For most models, the compilation of the stan code takes around 45 seconds.
day.form <- bf(BARNACLE ~ TREAT,
family = poisson(link = "log")
)
day.brm <- brm(day.form,
data = day,
iter = 5000,
warmup = 2500,
chains = 3, cores = 3,
thin = 5,
refresh = 0,
backend = "rstan"
)Compiling Stan program...Start sampling prior class coef group resp dpar nlpar lb ub source
(flat) b default
(flat) b TREATALG2 (vectorized)
(flat) b TREATNB (vectorized)
(flat) b TREATS (vectorized)
student_t(3, 3, 2.5) Intercept defaultThis tells us:
for the intercept, it is using a student t (flatter normal) prior with a mean of 0 and a standard deviation of 2.5. These mean and standard deviation values are the defaults.
for the beta coefficients (in this case, each effect), the default prior is a improper flat prior. A flat prior essentially means that any value between negative infinity and positive infinity are equally likely. Whilst this might seem reckless, in practice, it seems to work reasonably well for non-intercept beta parameters.
since we have nominated a Poisson distribution, there is no auxiliary prior.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Conclusions:
- we see that the range of predictions is fairly wide (ranging from 0 to very high)
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.
Data points may overlap. Use the `jitter` argument to add some amount of
random variation to the location of data points and avoid overplotting.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
[1] "b_Intercept" "b_TREATALG2" "b_TREATNB" "b_TREATS"
[5] "Intercept" "prior_Intercept" "prior_b" "lprior"
[9] "lp__" "accept_stat__" "stepsize__" "treedepth__"
[13] "n_leapfrog__" "divergent__" "energy__" 
$N
[1] 20
$Y
[1] 27 19 18 23 25 24 33 27 26 32 9 13 17 14 22 12 8 15 20 11
$K
[1] 4
$Kc
[1] 3
$X
Intercept TREATALG2 TREATNB TREATS
1 1 0 0 0
2 1 0 0 0
3 1 0 0 0
4 1 0 0 0
5 1 0 0 0
6 1 1 0 0
7 1 1 0 0
8 1 1 0 0
9 1 1 0 0
10 1 1 0 0
11 1 0 1 0
12 1 0 1 0
13 1 0 1 0
14 1 0 1 0
15 1 0 1 0
16 1 0 0 1
17 1 0 0 1
18 1 0 0 1
19 1 0 0 1
20 1 0 0 1
attr(,"assign")
[1] 0 1 1 1
attr(,"contrasts")
attr(,"contrasts")$TREAT
ALG2 NB S
ALG1 0 0 0
ALG2 1 0 0
NB 0 1 0
S 0 0 1
$prior_only
[1] 0
attr(,"class")
[1] "standata" "list" // generated with brms 2.22.0
functions {
}
data {
int<lower=1> N; // total number of observations
array[N] int Y; // response variable
int<lower=1> K; // number of population-level effects
matrix[N, K] X; // population-level design matrix
int<lower=1> Kc; // number of population-level effects after centering
int prior_only; // should the likelihood be ignored?
}
transformed data {
matrix[N, Kc] Xc; // centered version of X without an intercept
vector[Kc] means_X; // column means of X before centering
for (i in 2:K) {
means_X[i - 1] = mean(X[, i]);
Xc[, i - 1] = X[, i] - means_X[i - 1];
}
}
parameters {
vector[Kc] b; // regression coefficients
real Intercept; // temporary intercept for centered predictors
}
transformed parameters {
real lprior = 0; // prior contributions to the log posterior
lprior += normal_lpdf(b | 0, 2.4);
lprior += normal_lpdf(Intercept | 3.1, 2.2);
}
model {
// likelihood including constants
if (!prior_only) {
target += poisson_log_glm_lpmf(Y | Xc, Intercept, b);
}
// priors including constants
target += lprior;
}
generated quantities {
// actual population-level intercept
real b_Intercept = Intercept - dot_product(means_X, b);
// additionally sample draws from priors
real prior_b = normal_rng(0,2.4);
real prior_Intercept = normal_rng(3.1,2.2);
}6 MCMC sampling diagnostics
The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.
See list of available diagnostics by name
bayesplot MCMC module:
mcmc_acf
mcmc_acf_bar
mcmc_areas
mcmc_areas_ridges
mcmc_combo
mcmc_dens
mcmc_dens_chains
mcmc_dens_overlay
mcmc_hex
mcmc_hist
mcmc_hist_by_chain
mcmc_intervals
mcmc_neff
mcmc_neff_hist
mcmc_nuts_acceptance
mcmc_nuts_divergence
mcmc_nuts_energy
mcmc_nuts_stepsize
mcmc_nuts_treedepth
mcmc_pairs
mcmc_parcoord
mcmc_rank_ecdf
mcmc_rank_hist
mcmc_rank_overlay
mcmc_recover_hist
mcmc_recover_intervals
mcmc_recover_scatter
mcmc_rhat
mcmc_rhat_hist
mcmc_scatter
mcmc_trace
mcmc_trace_highlight
mcmc_violinOf these, we will focus on:
- mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
There is no evidence of auto-correlation in the MCMC samples
- Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Ratios all very high.
The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
There is no evidence of auto-correlation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Ratios all very high.
The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
There is no evidence of auto-correlation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Warning: Returning more (or less) than 1 row per `summarise()` group was deprecated in
dplyr 1.1.0.
ℹ Please use `reframe()` instead.
ℹ When switching from `summarise()` to `reframe()`, remember that `reframe()`
always returns an ungrouped data frame and adjust accordingly.
ℹ The deprecated feature was likely used in the ggmcmc package.
Please report the issue at <https://github.com/xfim/ggmcmc/issues/>.
Ratios all very high.
6.0.1 stan plots
The brms package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- stan_acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
There is no evidence of auto-correlation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Ratios all very high.
7 Model validation
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
See list of available diagnostics by name
bayesplot PPC module:
ppc_bars
ppc_bars_grouped
ppc_boxplot
ppc_dens
ppc_dens_overlay
ppc_dens_overlay_grouped
ppc_ecdf_overlay
ppc_ecdf_overlay_grouped
ppc_error_binned
ppc_error_hist
ppc_error_hist_grouped
ppc_error_scatter
ppc_error_scatter_avg
ppc_error_scatter_avg_grouped
ppc_error_scatter_avg_vs_x
ppc_freqpoly
ppc_freqpoly_grouped
ppc_hist
ppc_intervals
ppc_intervals_grouped
ppc_km_overlay
ppc_km_overlay_grouped
ppc_loo_intervals
ppc_loo_pit_ecdf
ppc_loo_pit_overlay
ppc_loo_pit_qq
ppc_loo_ribbon
ppc_pit_ecdf
ppc_pit_ecdf_grouped
ppc_ribbon
ppc_ribbon_grouped
ppc_rootogram
ppc_scatter
ppc_scatter_avg
ppc_scatter_avg_grouped
ppc_stat
ppc_stat_2d
ppc_stat_freqpoly
ppc_stat_freqpoly_grouped
ppc_stat_grouped
ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
The model draws appear deviate from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.
- error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.
- intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
The modelled predictions seem to underestimate the uncertainty with increasing mussel clump area.
The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.
DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- posterior_predict(day.rstanarm3, ndraws = 250, summary = FALSE)
day.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = day$BARNACLE,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
plot(day.resids)
Conclusions:
- the simulated residuals DOES NOT suggest any issues with the fitted model
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
See list of available diagnostics by name
bayesplot PPC module:
ppc_bars
ppc_bars_grouped
ppc_boxplot
ppc_dens
ppc_dens_overlay
ppc_dens_overlay_grouped
ppc_ecdf_overlay
ppc_ecdf_overlay_grouped
ppc_error_binned
ppc_error_hist
ppc_error_hist_grouped
ppc_error_scatter
ppc_error_scatter_avg
ppc_error_scatter_avg_grouped
ppc_error_scatter_avg_vs_x
ppc_freqpoly
ppc_freqpoly_grouped
ppc_hist
ppc_intervals
ppc_intervals_grouped
ppc_km_overlay
ppc_km_overlay_grouped
ppc_loo_intervals
ppc_loo_pit_ecdf
ppc_loo_pit_overlay
ppc_loo_pit_qq
ppc_loo_ribbon
ppc_pit_ecdf
ppc_pit_ecdf_grouped
ppc_ribbon
ppc_ribbon_grouped
ppc_rootogram
ppc_scatter
ppc_scatter_avg
ppc_scatter_avg_grouped
ppc_stat
ppc_stat_2d
ppc_stat_freqpoly
ppc_stat_freqpoly_grouped
ppc_stat_grouped
ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
The model draws appear deviate from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
Using all posterior draws for ppc type 'error_scatter_avg' by default.
The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.
- intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
Using all posterior draws for ppc type 'intervals' by default.Warning: The following arguments were unrecognized and ignored: group
The modelled predictions seem to underestimate the uncertainty with increasing mussel clump area.
The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.
DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- day.brm3 |> posterior_predict(ndraws = 250, summary = FALSE)
day.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = day$BARNACLE,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
day.resids |> plot()
day.resids <- make_brms_dharma_res(day.brm3, integerResponse = TRUE)
wrap_elements(~ testUniformity(day.resids)) +
wrap_elements(~ plotResiduals(day.resids, form = factor(rep(1, nrow(day))))) +
wrap_elements(~ plotResiduals(day.resids, quantreg = TRUE)) +
wrap_elements(~ testDispersion(day.resids))
Conclusions:
- the simulated residuals DO NOT suggest any issues with the model fit
8 Partial effects plots
9 Model investigation
The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).
Model Info:
function: stan_glm
family: poisson [log]
formula: BARNACLE ~ TREAT
algorithm: sampling
sample: 2400 (posterior sample size)
priors: see help('prior_summary')
observations: 20
predictors: 4
Estimates:
mean sd 10% 50% 90%
(Intercept) 3.1 0.1 3.0 3.1 3.2
TREATALG2 0.2 0.1 0.1 0.2 0.4
TREATNB -0.4 0.1 -0.6 -0.4 -0.2
TREATS -0.5 0.2 -0.7 -0.5 -0.3
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 19.7 1.4 18.0 19.7 21.5
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 2455
TREATALG2 0.0 1.0 2444
TREATNB 0.0 1.0 2429
TREATS 0.0 1.0 2284
mean_PPD 0.0 1.0 2357
log-posterior 0.0 1.0 2305
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).Conclusions:
- in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 20 raw observations.
- the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is 3.1. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 22.25.
- the estimated effect of ALG2 vs ALG1 is 0.23 (mean) or 0.07 (median) with a standard deviation of 0. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.23 and 0.24 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 1.26 times higher than that on ALG1. This represents a 26% increase in barnacle recruitment.
- the estimated effect of NB and S are -0.39 and -0.53 respectively, which equate to 1.48 and 1.69 fold declines respectively.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
tidyMCMC(day.rstanarm3$stanfit, estimate.method = "median", conf.int = TRUE, conf.method = "HPDinterval", rhat = TRUE, ess = TRUE)# A tibble: 6 × 7
term estimate std.error conf.low conf.high rhat ess
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 (Intercept) 3.10 0.0930 2.93 3.28 1.00 2455
2 TREATALG2 0.235 0.126 0.0171 0.506 1.000 2444
3 TREATNB -0.393 0.146 -0.666 -0.105 0.999 2429
4 TREATS -0.526 0.154 -0.810 -0.220 0.999 2284
5 mean_PPD 19.7 1.39 17.0 22.4 1.000 2357
6 log-posterior -62.1 1.38 -64.8 -60.2 1.00 2305Conclusions:
- the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is 3.1. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 22.25.
- the estimated effect of ALG2 vs ALG1 is 0.23 (median) with a standard error of 0.13. The 95% credibility intervals indicate that we are 95% confident that the effect is between 0.02 and 0.51 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 1.26 times higher than that on ALG1. This represents a 26% increase in barnacle recruitment.
- the estimated effect of NB and S are -0.39 and -0.53 respectively, which equate to 1.48 and 1.69 fold declines respectively.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
day.rstanarm3$stanfit |>
summarise_draws(
median,
HDInterval::hdi,
rhat, length, ess_bulk, ess_tail
)# A tibble: 6 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 3.10 2.93 3.28 1.00 2400 2464. 2266.
2 TREATALG2 0.238 0.0171 0.506 1.000 2400 2454. 2180.
3 TREATNB -0.395 -0.666 -0.105 1.000 2400 2432. 2410.
4 TREATS -0.526 -0.810 -0.220 1.000 2400 2286. 2354.
5 mean_PPD 19.7 17.0 22.4 1.00 2400 2360. 2130.
6 log-posterior -61.8 -64.8 -60.2 1.00 2400 2340. 2250.We can also alter the CI level.
day.rstanarm3$stanfit |>
summarise_draws(
median,
~ HDInterval::hdi(.x, credMass = 0.9),
rhat, length, ess_bulk, ess_tail
)# A tibble: 6 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 3.10 2.95 3.25 1.00 2400 2464. 2266.
2 TREATALG2 0.238 0.0237 0.436 1.000 2400 2454. 2180.
3 TREATNB -0.395 -0.631 -0.154 1.000 2400 2432. 2410.
4 TREATS -0.526 -0.765 -0.261 1.000 2400 2286. 2354.
5 mean_PPD 19.7 17.2 21.7 1.00 2400 2360. 2130.
6 log-posterior -61.8 -64.0 -60.2 1.00 2400 2340. 2250.Arguably, it would be better to back-transform to the ratio scale
day.rstanarm3$stanfit |>
summarise_draws(
~ median(exp(.x)),
~ HDInterval::hdi(exp(.x)),
rhat, length, ess_bulk, ess_tail
)# A tibble: 6 × 8
variable `~median(exp(.x))` lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 (Intercep… 2.22e+ 1 1.83e+ 1 2.63e+ 1 1.00 2400 2464. 2266.
2 TREATALG2 1.27e+ 0 9.75e- 1 1.59e+ 0 1.000 2400 2454. 2180.
3 TREATNB 6.74e- 1 5.10e- 1 8.95e- 1 1.000 2400 2432. 2410.
4 TREATS 5.91e- 1 4.34e- 1 7.87e- 1 1.000 2400 2286. 2354.
5 mean_PPD 3.59e+ 8 8.04e+ 6 4.17e+ 9 1.00 2400 2360. 2130.
6 log-poste… 1.52e-27 1.90e-30 5.40e-27 1.00 2400 2340. 2250.# A draws_df: 800 iterations, 3 chains, and 6 variables
(Intercept) TREATALG2 TREATNB TREATS mean_PPD log-posterior
1 3.2 0.030 -0.531 -0.52 20 -62
2 3.2 0.048 -0.404 -0.64 21 -62
3 3.0 0.398 -0.234 -0.33 20 -61
4 3.3 0.089 -0.291 -0.78 22 -66
5 3.2 0.298 -0.237 -0.57 23 -65
6 3.1 0.204 -0.591 -0.64 18 -61
7 2.9 0.196 -0.084 -0.31 18 -67
8 3.0 0.486 -0.117 -0.48 20 -63
9 3.1 0.169 -0.455 -0.49 20 -60
10 3.1 0.150 -0.586 -0.42 18 -63
# ... with 2390 more draws
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}day.rstanarm3$stanfit |>
as_draws_df() |>
summarise_draws(
median,
~ HDInterval::hdi(.x),
rhat,
length,
ess_bulk, ess_tail
)# A tibble: 6 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 3.10 2.93 3.28 1.00 2400 2464. 2266.
2 TREATALG2 0.238 0.0171 0.506 1.000 2400 2454. 2180.
3 TREATNB -0.395 -0.666 -0.105 1.000 2400 2432. 2410.
4 TREATS -0.526 -0.810 -0.220 1.000 2400 2286. 2354.
5 mean_PPD 19.7 17.0 22.4 1.00 2400 2360. 2130.
6 log-posterior -61.8 -64.8 -60.2 1.00 2400 2340. 2250.day.rstanarm3$stanfit |>
as_draws_df() |>
exp() |>
summarise_draws(
median,
~ HDInterval::hdi(.x),
rhat,
length,
ess_bulk, ess_tail
)# A tibble: 6 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 2.22e+ 1 1.83e+ 1 2.63e+ 1 1.00 2400 2464. 2266.
2 TREATALG2 1.27e+ 0 9.75e- 1 1.59e+ 0 1.000 2400 2454. 2180.
3 TREATNB 6.74e- 1 5.10e- 1 8.95e- 1 1.000 2400 2432. 2410.
4 TREATS 5.91e- 1 4.34e- 1 7.87e- 1 1.000 2400 2286. 2354.
5 mean_PPD 3.59e+ 8 8.04e+ 6 4.17e+ 9 1.00 2400 2360. 2130.
6 log-posterior 1.52e-27 1.90e-30 5.40e-27 1.00 2400 2340. 2250.Due to the presence of a log transform in the predictor, it is better to use the regex version.
[1] "(Intercept)" "TREATALG2" "TREATNB" "TREATS"
[5] "accept_stat__" "stepsize__" "treedepth__" "n_leapfrog__"
[9] "divergent__" "energy__" # A tibble: 9,600 × 5
# Groups: .variable [4]
.chain .iteration .draw .variable .value
<int> <int> <int> <chr> <dbl>
1 1 1 1 (Intercept) 3.20
2 1 2 2 (Intercept) 3.25
3 1 3 3 (Intercept) 3.00
4 1 4 4 (Intercept) 3.28
5 1 5 5 (Intercept) 3.20
6 1 6 6 (Intercept) 3.15
7 1 7 7 (Intercept) 2.89
8 1 8 8 (Intercept) 2.96
9 1 9 9 (Intercept) 3.14
10 1 10 10 (Intercept) 3.07
# ℹ 9,590 more rowsexceedP <- function(x, Val = 0) mean(x > Val)
day.rstanarm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
mutate(.value = exp(.value)) |>
summarise_draws(
median,
HDInterval::hdi,
rhat,
length,
ess_bulk,
ess_tail,
~ exceedP(.x, 1)
)# A tibble: 4 × 10
# Groups: .variable [4]
.variable variable median lower upper rhat length ess_bulk ess_tail
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) .value 22.2 18.3 26.3 1.00 2400 2464. 2266.
2 TREATALG2 .value 1.27 0.975 1.59 1.000 2400 2454. 2180.
3 TREATNB .value 0.674 0.510 0.895 1.000 2400 2432. 2410.
4 TREATS .value 0.591 0.434 0.787 1.000 2400 2286. 2354.
# ℹ 1 more variable: `~exceedP(.x, 1)` <dbl>We can then summarise this
# A tibble: 4 × 7
.variable .value .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 (Intercept) 3.10 2.93 3.28 0.95 median hdci
2 TREATALG2 0.238 -0.0258 0.465 0.95 median hdci
3 TREATNB -0.395 -0.672 -0.111 0.95 median hdci
4 TREATS -0.526 -0.810 -0.220 0.95 median hdci We could alternatively express the parameters on the response scale.
# A tibble: 4 × 7
.variable `exp(.value)` .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 (Intercept) 22.2 18.3 26.4 0.95 median hdci
2 TREATALG2 1.27 0.968 1.59 0.95 median hdci
3 TREATNB 0.674 0.510 0.895 0.95 median hdci
4 TREATS 0.591 0.434 0.787 0.95 median hdci Conclusions:
- the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is 3.1. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 22.23.
- the estimated effect of ALG2 vs ALG1 is 0.24 (median) with a standard error of -0.03. The 95% credibility intervals indicate that we are 95% confident that the effect is between 0.46 and 0.95 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 1.27 times higher than that on ALG1. This represents a 27% increase in barnacle recruitment.
- the estimated effect of NB and S are -0.4 and -0.53 respectively, which equate to 1.48 and 1.69 fold declines respectively.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
day.rstanarm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = .value, y = .variable)) +
facet_wrap(~.variable, scales = "free")
day.rstanarm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = .value, y = .variable)) +
geom_vline(xintercept = 0, linetype = "dashed")
day.rstanarm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = exp(.value), y = .variable)) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans())
day.rstanarm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
geom_density_ridges(aes(x = .value, y = .variable), alpha = 0.4) +
geom_vline(xintercept = 0, linetype = "dashed")Picking joint bandwidth of 0.0269
## Or on a fractional scale
day.rstanarm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
geom_density_ridges_gradient(
aes(
x = exp(.value),
y = .variable,
fill = stat(x)
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_c(option = "C")Warning: `stat(x)` was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(x)` instead.Picking joint bandwidth of 0.0388
This is purely a graphical depiction on the posteriors.
# A tibble: 2,400 × 13
.chain .iteration .draw `(Intercept)` TREATALG2 TREATNB TREATS accept_stat__
<int> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
1 1 1 1 3.20 0.0296 -0.531 -0.520 0.985
2 1 2 2 3.25 0.0477 -0.404 -0.643 0.972
3 1 3 3 3.00 0.398 -0.234 -0.325 0.981
4 1 4 4 3.28 0.0885 -0.291 -0.784 0.848
5 1 5 5 3.20 0.298 -0.237 -0.566 0.952
6 1 6 6 3.15 0.204 -0.591 -0.644 0.997
7 1 7 7 2.89 0.196 -0.0843 -0.311 0.736
8 1 8 8 2.96 0.486 -0.117 -0.479 0.986
9 1 9 9 3.14 0.169 -0.455 -0.489 0.986
10 1 10 10 3.07 0.150 -0.586 -0.420 0.995
# ℹ 2,390 more rows
# ℹ 5 more variables: stepsize__ <dbl>, treedepth__ <dbl>, n_leapfrog__ <dbl>,
# divergent__ <dbl>, energy__ <dbl># A tibble: 2,400 × 7
.chain .iteration .draw `(Intercept)` TREATALG2 TREATNB TREATS
<int> <int> <int> <dbl> <dbl> <dbl> <dbl>
1 1 1 1 3.20 0.0296 -0.531 -0.520
2 1 2 2 3.25 0.0477 -0.404 -0.643
3 1 3 3 3.00 0.398 -0.234 -0.325
4 1 4 4 3.28 0.0885 -0.291 -0.784
5 1 5 5 3.20 0.298 -0.237 -0.566
6 1 6 6 3.15 0.204 -0.591 -0.644
7 1 7 7 2.89 0.196 -0.0843 -0.311
8 1 8 8 2.96 0.486 -0.117 -0.479
9 1 9 9 3.14 0.169 -0.455 -0.489
10 1 10 10 3.07 0.150 -0.586 -0.420
# ℹ 2,390 more rowsWarning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.# A tibble: 2,400 × 4
`(Intercept)` TREATALG2 TREATNB TREATS
<dbl> <dbl> <dbl> <dbl>
1 3.20 0.0296 -0.531 -0.520
2 3.25 0.0477 -0.404 -0.643
3 3.00 0.398 -0.234 -0.325
4 3.28 0.0885 -0.291 -0.784
5 3.20 0.298 -0.237 -0.566
6 3.15 0.204 -0.591 -0.644
7 2.89 0.196 -0.0843 -0.311
8 2.96 0.486 -0.117 -0.479
9 3.14 0.169 -0.455 -0.489
10 3.07 0.150 -0.586 -0.420
# ℹ 2,390 more rowsThe summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).
Family: poisson
Links: mu = log
Formula: BARNACLE ~ TREAT
Data: day (Number of observations: 20)
Draws: 3 chains, each with iter = 5000; warmup = 1000; thin = 5;
total post-warmup draws = 2400
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 3.10 0.09 2.91 3.28 1.00 2423 2253
TREATALG2 0.24 0.12 -0.00 0.49 1.00 2415 2290
TREATNB -0.40 0.15 -0.71 -0.11 1.00 2263 2081
TREATS -0.53 0.16 -0.83 -0.23 1.00 2401 2455
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Conclusions:
- in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 20 raw observations.
- the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is 3.1. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 22.28.
- the estimated effect of ALG2 vs ALG1 is 0.24 (mean) or 0.49 (median) with a standard deviation of 0.12. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.24 and 1 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 1.27 times higher than that on ALG1. This represents a 27% increase in barnacle recruitment.
- the estimated effect of NB and S are -0.4 and -0.53 respectively, which equate to 1.5 and 1.69 fold decline respectively.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
day.brm3$fit |> tidyMCMC(
estimate.method = "median",
conf.int = TRUE,
conf.method = "HPDinterval",
rhat = TRUE,
ess = TRUE
)# A tibble: 8 × 7
term estimate std.error conf.low conf.high rhat ess
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 b_Intercept 3.10 0.0946 2.92 3.29 0.999 2416
2 b_TREATALG2 0.242 0.124 0.0168 0.501 1.00 2406
3 b_TREATNB -0.405 0.150 -0.721 -0.124 0.999 2264
4 b_TREATS -0.527 0.158 -0.817 -0.222 1.000 2401
5 Intercept 2.93 0.0533 2.83 3.04 1.000 2460
6 prior_Intercept 3.10 2.12 -0.984 7.36 1.00 2317
7 prior_b 0.000881 2.43 -4.83 4.71 1.000 2293
8 lprior -7.14 0.0201 -7.18 -7.11 0.999 2356Conclusions:
- the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is 3.1. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 22.28.
- the estimated effect of ALG2 vs ALG1 is 0.24 (median) with a standard error of 0.12. The 95% credibility intervals indicate that we are 95% confident that the effect is between 0.02 and 0.5 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 1.27 times higher than that on ALG1. This represents a 27% increase in barnacle recruitment.
- the estimated effect of NB and S are -0.4 and -0.53 respectively, which equate to 1.5 and 1.69 fold declines respectively.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
# A draws_df: 800 iterations, 3 chains, and 9 variables
b_Intercept b_TREATALG2 b_TREATNB b_TREATS Intercept prior_Intercept prior_b
1 3.0 0.14 -0.30 -0.45 2.9 6.44 1.20
2 3.2 0.14 -0.48 -0.66 2.9 0.62 -1.91
3 3.1 0.22 -0.39 -0.20 3.0 1.90 1.58
4 3.2 0.22 -0.47 -0.75 2.9 6.31 1.64
5 3.1 0.31 -0.31 -1.00 2.8 1.63 -3.95
6 3.0 0.35 -0.13 -0.48 2.9 2.52 0.97
7 3.1 0.24 -0.37 -0.51 3.0 3.81 -1.52
8 3.1 0.35 -0.57 -0.50 2.9 8.28 3.94
9 3.1 0.29 -0.24 -0.51 3.0 3.24 0.89
10 3.1 0.47 -0.44 -0.74 3.0 2.66 -2.22
lprior
1 -7.1
2 -7.2
3 -7.1
4 -7.2
5 -7.2
6 -7.1
7 -7.1
8 -7.2
9 -7.1
10 -7.2
# ... with 2390 more draws, and 1 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}day.brm3 |>
as_draws_df() |>
summarise_draws(
median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail
)# A tibble: 9 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 b_Intercept 3.11 2.92 3.29 0.999 2400 2423. 2253.
2 b_TREATALG2 0.241 0.0168 0.501 1.00 2400 2415. 2290.
3 b_TREATNB -0.405 -0.721 -0.124 1.000 2400 2263. 2081.
4 b_TREATS -0.524 -0.817 -0.222 1.000 2400 2401. 2455.
5 Intercept 2.93 2.83 3.04 1.000 2400 2469. 2367.
6 prior_Intercept 3.11 -0.984 7.36 1.00 2400 2321. 2314.
7 prior_b -0.0915 -4.83 4.71 1.000 2400 2293. 2290.
8 lprior -7.14 -7.18 -7.11 1.00 2400 2317. 1944.
9 lp__ -65.0 -68.1 -63.3 1.000 2400 2477. 2291.day.brm3 |>
as_draws_df() |>
exp() |>
summarise_draws(
median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail
)# A tibble: 9 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 b_Intercept 2.23e+ 1 1.85e+ 1 2.67e+ 1 0.999 2400 2423. 2253.
2 b_TREATALG2 1.27e+ 0 1.00e+ 0 1.63e+ 0 1.00 2400 2415. 2290.
3 b_TREATNB 6.67e- 1 4.69e- 1 8.61e- 1 1.000 2400 2263. 2081.
4 b_TREATS 5.92e- 1 4.33e- 1 7.91e- 1 1.000 2400 2401. 2455.
5 Intercept 1.88e+ 1 1.68e+ 1 2.07e+ 1 1.000 2400 2469. 2367.
6 prior_Intercept 2.24e+ 1 1.53e- 2 6.66e+ 2 1.00 2400 2321. 2314.
7 prior_b 9.13e- 1 4.37e- 4 5.44e+ 1 1.000 2400 2293. 2290.
8 lprior 7.93e- 4 7.59e- 4 8.18e- 4 1.00 2400 2317. 1944.
9 lp__ 6.11e-29 9.06e-33 2.20e-28 1.000 2400 2477. 2291.Due to the presence of a log transform in the predictor, it is better to use the regex version.
[1] "b_Intercept" "b_TREATALG2" "b_TREATNB" "b_TREATS"
[5] "Intercept" "prior_Intercept" "prior_b" "lprior"
[9] "lp__" "accept_stat__" "stepsize__" "treedepth__"
[13] "n_leapfrog__" "divergent__" "energy__" # A tibble: 7,200 × 5
# Groups: .variable [3]
.chain .iteration .draw .variable .value
<int> <int> <int> <chr> <dbl>
1 1 1 1 b_TREATALG2 0.143
2 1 2 2 b_TREATALG2 0.140
3 1 3 3 b_TREATALG2 0.216
4 1 4 4 b_TREATALG2 0.217
5 1 5 5 b_TREATALG2 0.310
6 1 6 6 b_TREATALG2 0.352
7 1 7 7 b_TREATALG2 0.236
8 1 8 8 b_TREATALG2 0.350
9 1 9 9 b_TREATALG2 0.290
10 1 10 10 b_TREATALG2 0.474
# ℹ 7,190 more rowsday.brm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
mutate(.value = exp(.value)) |>
summarise_draws(
median,
~ HDInterval::hdi(.x, credMass = 0.95),
rhat,
length,
ess_bulk, ess_tail
)# A tibble: 3 × 9
# Groups: .variable [3]
.variable variable median lower upper rhat length ess_bulk ess_tail
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 b_TREATALG2 .value 1.27 1.00 1.63 1.00 2400 2415. 2290.
2 b_TREATNB .value 0.667 0.469 0.861 1.000 2400 2263. 2081.
3 b_TREATS .value 0.592 0.433 0.791 1.000 2400 2401. 2455.exceedP <- function(x, Val = 0) mean(x > Val)
day.brm3 |>
tidy_draws() |>
exp() |>
dplyr::select(starts_with("b_")) |>
summarise_draws(
median,
~ HDInterval::hdi(.x, credMass = 0.9),
rhat,
ess_bulk, ess_tail,
~ exceedP(.x, 1)
)# A tibble: 4 × 8
variable median lower upper rhat ess_bulk ess_tail `~exceedP(.x, 1)`
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 b_Intercept 22.3 18.8 25.8 1.000 2412. 2249. 1
2 b_TREATALG2 1.27 1.02 1.53 1.000 2404. 2281. 0.974
3 b_TREATNB 0.667 0.519 0.844 1.000 2253. 2053. 0.00667
4 b_TREATS 0.592 0.442 0.749 1.000 2390. 2447. 0 We can then summarise this
# A tibble: 3 × 7
.variable .value .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 b_TREATALG2 0.241 0.0168 0.501 0.95 median hdci
2 b_TREATNB -0.405 -0.719 -0.121 0.95 median hdci
3 b_TREATS -0.524 -0.817 -0.222 0.95 median hdci We could alternatively express the parameters on the response scale.
# A tibble: 3 × 7
.variable `exp(.value)` .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 b_TREATALG2 1.27 0.982 1.61 0.95 median hdci
2 b_TREATNB 0.667 0.476 0.872 0.95 median hdci
3 b_TREATS 0.592 0.434 0.792 0.95 median hdci day.brm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = .value, y = .variable)) +
facet_wrap(~.variable, scales = "free")
Conclusions:
- the estimated mean (expected number of newly recruited barnacles) on the ALG1 surface is 0.24. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 1.27.
- the estimated effect of ALG2 vs ALG1 is -0.4 (median) with a standard error of -0.72. The 95% credibility intervals indicate that we are 95% confident that the effect is between -0.12 and 0.95 - e.g. there is a significant positive effect. When back-transformed onto the response scale, we see that barnacle recruitment on ALG2 is 0.67 times higher than that on ALG1. This represents a -33% increase in barnacle recruitment.
- the estimated effect of NB and S are -0.52 and NA respectively, which equate to 1.69 and NA fold declines respectively.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
## Link scale
day.brm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_slab(aes(
x = .value, y = .variable,
fill = stat(ggdist::cut_cdf_qi(cdf,
.width = c(0.5, 0.8, 0.95),
labels = scales::percent_format()
))
), color = "black") +
geom_vline(xintercept = 0, linetype = "dashed") +
scale_fill_brewer("Interval", direction = -1, na.translate = FALSE)
## Fractional scale
day.brm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
mutate(.value = exp(.value)) |>
ggplot() +
stat_slab(aes(
x = .value, y = .variable,
fill = stat(ggdist::cut_cdf_qi(cdf,
.width = c(0.5, 0.8, 0.95),
labels = scales::percent_format()
))
), color = "black") +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_fill_brewer("Interval", direction = -1, na.translate = FALSE) +
scale_x_continuous(trans = scales::log2_trans())
day.brm3 |>
gather_draws(`.Intercept.*|.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = .value, y = .variable)) +
facet_wrap(~.variable, scales = "free")
day.brm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = .value, y = .variable)) +
geom_vline(xintercept = 0, linetype = "dashed")
day.brm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
stat_halfeye(aes(x = exp(.value), y = .variable)) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans())
day.brm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
geom_density_ridges(aes(x = .value, y = .variable), alpha = 0.4) +
geom_vline(xintercept = 0, linetype = "dashed")Picking joint bandwidth of 0.0269
## Or on a fractional scale
day.brm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
geom_density_ridges_gradient(
aes(
x = exp(.value),
y = .variable,
fill = stat(x)
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_c(option = "C")Picking joint bandwidth of 0.0388
This is purely a graphical depiction on the posteriors.
# A tibble: 2,400 × 18
.chain .iteration .draw b_Intercept b_TREATALG2 b_TREATNB b_TREATS Intercept
<int> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
1 1 1 1 3.04 0.143 -0.300 -0.451 2.89
2 1 2 2 3.18 0.140 -0.478 -0.664 2.93
3 1 3 3 3.07 0.216 -0.393 -0.197 2.97
4 1 4 4 3.17 0.217 -0.468 -0.748 2.92
5 1 5 5 3.10 0.310 -0.314 -0.999 2.84
6 1 6 6 2.99 0.352 -0.127 -0.479 2.92
7 1 7 7 3.12 0.236 -0.371 -0.505 2.96
8 1 8 8 3.06 0.350 -0.573 -0.500 2.88
9 1 9 9 3.08 0.290 -0.236 -0.513 2.96
10 1 10 10 3.15 0.474 -0.437 -0.738 2.97
# ℹ 2,390 more rows
# ℹ 10 more variables: prior_Intercept <dbl>, prior_b <dbl>, lprior <dbl>,
# lp__ <dbl>, accept_stat__ <dbl>, stepsize__ <dbl>, treedepth__ <dbl>,
# n_leapfrog__ <dbl>, divergent__ <dbl>, energy__ <dbl># A tibble: 2,400 × 6
.chain .iteration .draw b_TREATALG2 b_TREATNB b_TREATS
<int> <int> <int> <dbl> <dbl> <dbl>
1 1 1 1 0.143 -0.300 -0.451
2 1 2 2 0.140 -0.478 -0.664
3 1 3 3 0.216 -0.393 -0.197
4 1 4 4 0.217 -0.468 -0.748
5 1 5 5 0.310 -0.314 -0.999
6 1 6 6 0.352 -0.127 -0.479
7 1 7 7 0.236 -0.371 -0.505
8 1 8 8 0.350 -0.573 -0.500
9 1 9 9 0.290 -0.236 -0.513
10 1 10 10 0.474 -0.437 -0.738
# ℹ 2,390 more rowsWarning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.# A tibble: 2,400 × 9
b_Intercept b_TREATALG2 b_TREATNB b_TREATS Intercept prior_Intercept prior_b
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 3.04 0.143 -0.300 -0.451 2.89 6.44 1.20
2 3.18 0.140 -0.478 -0.664 2.93 0.617 -1.91
3 3.07 0.216 -0.393 -0.197 2.97 1.90 1.58
4 3.17 0.217 -0.468 -0.748 2.92 6.31 1.64
5 3.10 0.310 -0.314 -0.999 2.84 1.63 -3.95
6 2.99 0.352 -0.127 -0.479 2.92 2.52 0.970
7 3.12 0.236 -0.371 -0.505 2.96 3.81 -1.52
8 3.06 0.350 -0.573 -0.500 2.88 8.28 3.94
9 3.08 0.290 -0.236 -0.513 2.96 3.24 0.890
10 3.15 0.474 -0.437 -0.738 2.97 2.66 -2.22
# ℹ 2,390 more rows
# ℹ 2 more variables: lprior <dbl>, lp__ <dbl>Region of Practical Equivalence
[1] 0.04111272# Proportion of samples inside the ROPE [-0.04, 0.04]:
Parameter | inside ROPE
-----------------------
Intercept | 0.00 %
TREATALG2 | 2.94 %
TREATNB | 0.00 %
TREATS | 0.00 %
## Or based on fractional scale
day.brm3 |>
as_draws_df("^b_TREAT.*", regex = TRUE) |>
exp() |>
## equivalence_test(range = c(0.9, 1.1))
rope(range = c(0.9, 1.1))# Proportion of samples inside the ROPE [0.90, 1.10]:
Parameter | inside ROPE
-----------------------
TREATALG2 | 8.86 %
TREATNB | 0.00 %
TREATS | 0.00 %day.mcmc <-
day.brm3 |>
as_draws_df("^b_TREAT.*", regex = TRUE) |>
exp()
day.mcmc |>
rope(range = c(0.9, 1.1))# Proportion of samples inside the ROPE [0.90, 1.10]:
Parameter | inside ROPE
-----------------------
TREATALG2 | 8.86 %
TREATNB | 0.00 %
TREATS | 0.00 %
# Test for Practical Equivalence
ROPE: [0.90 1.10]
Parameter | H0 | inside ROPE | 95% HDI
--------------------------------------------------
TREATALG2 | Undecided | 8.86 % | [1.00, 1.63]
TREATNB | Rejected | 0.00 % | [0.49, 0.90]
TREATS | Rejected | 0.00 % | [0.44, 0.79]10 Further investigations
$emmeans
TREAT rate lower.HPD upper.HPD
ALG1 22.2 18.3 26.3
ALG2 28.2 23.5 32.9
NB 15.0 11.9 18.6
S 13.2 10.1 16.3
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95
$contrasts
contrast ratio lower.HPD upper.HPD
ALG1 / ALG2 0.788 0.603 0.983
ALG1 / NB 1.485 1.087 1.923
ALG1 / S 1.692 1.241 2.242
ALG2 / NB 1.880 1.408 2.434
ALG2 / S 2.145 1.601 2.869
NB / S 1.143 0.797 1.550
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 Conclusions:
- the contrasts section of the output indicates that there is evidence of:
- ALG1 has 1.48 fold (48%) more newly recruited barnacles than the NB substrate
- ALG1 has 1.69 fold (69%) more newly recruited barnacles than the S substrate
- ALG2 has 1.88 fold (88%) more newly recruited barnacles than the NB substrate
- ALG2 has 2.14 fold (114%) more newly recruited barnacles than the S substrate
- ALG1 was not found to be different to ALG2 and NB was not found to be different to S
day.em <- emmeans(day.rstanarm3, pairwise ~ TREAT, type = "link")$contrasts |>
gather_emmeans_draws() |>
mutate(Fit = exp(.value))
day.em |> head()# A tibble: 6 × 6
# Groups: contrast [1]
contrast .chain .iteration .draw .value Fit
<chr> <int> <int> <int> <dbl> <dbl>
1 ALG1 - ALG2 NA NA 1 -0.0296 0.971
2 ALG1 - ALG2 NA NA 2 -0.0477 0.953
3 ALG1 - ALG2 NA NA 3 -0.398 0.672
4 ALG1 - ALG2 NA NA 4 -0.0885 0.915
5 ALG1 - ALG2 NA NA 5 -0.298 0.743
6 ALG1 - ALG2 NA NA 6 -0.204 0.815day.em |>
group_by(contrast) |>
ggplot(aes(x = Fit)) +
geom_histogram() +
geom_vline(xintercept = 1, color = "red") +
facet_wrap(~contrast, scales = "free")`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
day.em |>
group_by(contrast) |>
ggplot(aes(x = Fit)) +
geom_density_ridges_gradient(aes(y = contrast, fill = stat(x)),
alpha = 0.4, color = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_fill_viridis_c(option = "C") +
scale_x_continuous(trans = scales::log2_trans())Picking joint bandwidth of 0.0403
day.em |>
ggplot(aes(x = Fit)) +
geom_density_ridges_gradient(
aes(
y = contrast,
fill = factor(stat(x > 0))
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_d()Picking joint bandwidth of 0.0403
# A tibble: 6 × 10
contrast .value .value.lower .value.upper Fit Fit.lower Fit.upper .width
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 ALG1 - ALG2 -0.238 -0.465 0.0258 0.788 0.603 0.983 0.95
2 ALG1 - NB 0.395 0.111 0.672 1.48 1.10 1.94 0.95
3 ALG1 - S 0.526 0.220 0.810 1.69 1.24 2.24 0.95
4 ALG2 - NB 0.631 0.367 0.910 1.88 1.40 2.43 0.95
5 ALG2 - S 0.763 0.471 1.05 2.14 1.58 2.85 0.95
6 NB - S 0.133 -0.184 0.466 1.14 0.797 1.55 0.95
# ℹ 2 more variables: .point <chr>, .interval <chr># A tibble: 6 × 2
contrast P
<chr> <dbl>
1 ALG1 - ALG2 0.0338
2 ALG1 - NB 0.996
3 ALG1 - S 1
4 ALG2 - NB 1
5 ALG2 - S 1
6 NB - S 0.783 # A tibble: 6 × 2
contrast P
<chr> <dbl>
1 ALG1 - ALG2 0.966
2 ALG1 - NB 0.00417
3 ALG1 - S 0
4 ALG2 - NB 0
5 ALG2 - S 0
6 NB - S 0.217 ## Probability of effect greater than 10%
day.em |>
group_by(contrast) |>
summarize(P = mean(Fit > 1.1))# A tibble: 6 × 2
contrast P
<chr> <dbl>
1 ALG1 - ALG2 0.00708
2 ALG1 - NB 0.978
3 ALG1 - S 0.996
4 ALG2 - NB 1
5 ALG2 - S 1
6 NB - S 0.586 Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead. contrast ratio lower.HPD upper.HPD
ALG1 / ALG2 0.786 0.606 0.983
ALG1 / NB 1.499 1.095 2.004
ALG1 / S 1.689 1.217 2.224
ALG2 / NB 1.909 1.413 2.496
ALG2 / S 2.155 1.615 2.881
NB / S 1.130 0.793 1.518
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.# A tibble: 6 × 2
variable median
<chr> <dbl>
1 contrast ALG1 - ALG2 0.786
2 contrast ALG1 - NB 1.50
3 contrast ALG1 - S 1.69
4 contrast ALG2 - NB 1.91
5 contrast ALG2 - S 2.15
6 contrast NB - S 1.13 ## OR
day.brm3 |>
emmeans(~TREAT) |>
pairs() |>
gather_emmeans_draws() |>
mutate(.ratio = exp(.value)) |>
## median_hdci(.ratio)
summarise(
median_hdci(.ratio),
P = mean(.ratio > 1),
ROPE = rope(.ratio, range = c(0.9, 1.1))$ROPE_Percentage
)Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.# A tibble: 6 × 9
contrast y ymin ymax .width .point .interval P ROPE
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr> <dbl> <dbl>
1 ALG1 - ALG2 0.786 0.602 0.981 0.95 median hdci 0.0258 0.105
2 ALG1 - NB 1.50 1.10 2.00 0.95 median hdci 0.993 0
3 ALG1 - S 1.69 1.25 2.26 0.95 median hdci 1 0
4 ALG2 - NB 1.91 1.40 2.49 0.95 median hdci 1 0
5 ALG2 - S 2.15 1.62 2.88 0.95 median hdci 1 0
6 NB - S 1.13 0.782 1.51 0.95 median hdci 0.762 0.371## summarise(across(c(.value, .ratio), c(median, HDInterval::hdi)))
## summarise(across(c(.value, .ratio), c(median_hdci)))
day.mcmc <-
day.brm3 |>
emmeans(~TREAT) |>
pairs() |>
tidy_draws() |>
dplyr::select(-.chain, -.iteration, -.draw) |>
exp()Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.
Conclusions:
- the contrasts section of the output indicates that there is evidence of:
- ALG1 has 1.5 fold (50%) more newly recruited barnacles than the NB substrate
- ALG1 has 1.69 fold (69%) more newly recruited barnacles than the S substrate
- ALG2 has 1.91 fold (91%) more newly recruited barnacles than the NB substrate
- ALG2 has 2.15 fold (115%) more newly recruited barnacles than the S substrate
- ALG1 was not found to be different to ALG2 and NB was not found to be different to S
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead. contrast ratio lower.HPD upper.HPD
ALG1 / ALG2 0.786 0.606 0.983
ALG1 / NB 1.499 1.095 2.004
ALG1 / S 1.689 1.217 2.224
ALG2 / NB 1.909 1.413 2.496
ALG2 / S 2.155 1.615 2.881
NB / S 1.130 0.793 1.518
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 day.em <- day.brm3 |>
emmeans(~TREAT, type = "link") |>
pairs() |>
gather_emmeans_draws() |>
mutate(Fit = exp(.value))Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.day.em |>
group_by(contrast) |>
ggplot(aes(x = Fit)) +
geom_density_ridges_gradient(aes(y = contrast, fill = stat(x)),
alpha = 0.4, color = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_fill_viridis_c(option = "C") +
scale_x_continuous(trans = scales::log2_trans())Picking joint bandwidth of 0.0404
day.em |>
ggplot(aes(x = Fit)) +
geom_density_ridges_gradient(
aes(
y = contrast,
fill = factor(stat(x > 0))
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_d() +
scale_x_continuous(trans = scales::log2_trans())Scale for x is already present.
Adding another scale for x, which will replace the existing scale.
Picking joint bandwidth of 0.0404
# A tibble: 6 × 6
# Groups: contrast [1]
contrast .chain .iteration .draw .value Fit
<chr> <int> <int> <int> <dbl> <dbl>
1 ALG1 - ALG2 NA NA 1 -0.143 0.867
2 ALG1 - ALG2 NA NA 2 -0.140 0.870
3 ALG1 - ALG2 NA NA 3 -0.216 0.806
4 ALG1 - ALG2 NA NA 4 -0.217 0.805
5 ALG1 - ALG2 NA NA 5 -0.310 0.733
6 ALG1 - ALG2 NA NA 6 -0.352 0.703day.em |>
group_by(contrast) |>
ggplot(aes(x = Fit)) +
## geom_histogram() +
geom_halfeyeh() +
geom_vline(xintercept = 1, color = "red") +
facet_wrap(~contrast, scales = "free")Warning: 'geom_halfeyeh' is deprecated.
Use 'stat_halfeye' instead.
See help("Deprecated") and help("tidybayes-deprecated").
# A tibble: 6 × 10
contrast .value .value.lower .value.upper Fit Fit.lower Fit.upper .width
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 ALG1 - ALG2 -0.241 -0.501 -0.0168 0.786 0.602 0.981 0.95
2 ALG1 - NB 0.405 0.121 0.719 1.50 1.10 2.00 0.95
3 ALG1 - S 0.524 0.222 0.817 1.69 1.25 2.26 0.95
4 ALG2 - NB 0.646 0.359 0.923 1.91 1.40 2.49 0.95
5 ALG2 - S 0.768 0.501 1.07 2.15 1.62 2.88 0.95
6 NB - S 0.123 -0.220 0.428 1.13 0.782 1.51 0.95
# ℹ 2 more variables: .point <chr>, .interval <chr># A tibble: 6 × 2
contrast P
<chr> <dbl>
1 ALG1 - ALG2 0.0258
2 ALG1 - NB 0.993
3 ALG1 - S 1
4 ALG2 - NB 1
5 ALG2 - S 1
6 NB - S 0.762 ## Probability of effect greater than 10%
day.em |>
group_by(contrast) |>
summarize(P = mean(Fit > 1.1))# A tibble: 6 × 2
contrast P
<chr> <dbl>
1 ALG1 - ALG2 0.00292
2 ALG1 - NB 0.979
3 ALG1 - S 0.998
4 ALG2 - NB 1
5 ALG2 - S 1
6 NB - S 0.561 ## Effect size on absolute scale
day.em <- day.brm3 |>
emmeans(~TREAT, type = "link") |>
regrid() |>
pairs() |>
gather_emmeans_draws() |>
median_hdci(.value)Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.day.brm3 |>
emmeans(~TREAT, type = "link") |>
regrid() |>
pairs() |>
gather_emmeans_draws() |>
group_by(contrast) |>
ggplot(aes(x = .value)) +
geom_halfeyeh() +
geom_vline(xintercept = 0, color = "red") +
facet_wrap(~contrast, scales = "free")Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.Warning: 'geom_halfeyeh' is deprecated.
Use 'stat_halfeye' instead.
See help("Deprecated") and help("tidybayes-deprecated").
day.brm3 |>
emmeans(~TREAT, type = "response") |>
pairs() |>
gather_emmeans_draws() |>
mutate(.value = exp(.value)) |>
ggplot(aes(x = .value)) +
geom_density_ridges_gradient(
aes(
y = contrast,
fill = factor(stat(x > 0))
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_d()Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.Picking joint bandwidth of 0.0404
Define your own
Compare:
- ALG1 vs ALG2
- NB vs S
- average of ALG1+ALG2 vs NB+S
| Levels | Alg1 vs Alg2 | NB vs S | Alg vs Bare |
|---|---|---|---|
| Alg1 | 1 | 0 | 0.5 |
| Alg2 | -1 | 0 | 0.5 |
| NB | 0 | 1 | -0.5 |
| S | 0 | -1 | -0.5 |
## Planned contrasts
cmat <- cbind(
"Alg2_Alg1" = c(-1, 1, 0, 0),
"NB_S" = c(0, 0, 1, -1),
"Alg_Bare" = c(0.5, 0.5, -0.5, -0.5),
"Alg_NB" = c(0.5, 0.5, -1, 0)
)
# On the link scale
emmeans(day.rstanarm3, ~TREAT, contr = list(TREAT = cmat), type = "link")$emmeans
TREAT emmean lower.HPD upper.HPD
ALG1 3.10 2.93 3.28
ALG2 3.34 3.17 3.51
NB 2.71 2.48 2.93
S 2.58 2.33 2.81
Point estimate displayed: median
Results are given on the log (not the response) scale.
HPD interval probability: 0.95
$contrasts
contrast estimate lower.HPD upper.HPD
TREAT.Alg2_Alg1 0.238 0.0171 0.506
TREAT.NB_S 0.133 -0.1863 0.463
TREAT.Alg_Bare 0.579 0.3767 0.778
TREAT.Alg_NB 0.512 0.2628 0.755
Point estimate displayed: median
Results are given on the log (not the response) scale.
HPD interval probability: 0.95 # On the response scale
emmeans(day.rstanarm3, ~TREAT, contr = list(TREAT = cmat), type = "response")$emmeans
TREAT rate lower.HPD upper.HPD
ALG1 22.2 18.3 26.3
ALG2 28.2 23.5 32.9
NB 15.0 11.9 18.6
S 13.2 10.1 16.3
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95
$contrasts
contrast ratio lower.HPD upper.HPD
TREAT.Alg2_Alg1 1.27 0.975 1.59
TREAT.NB_S 1.14 0.797 1.55
TREAT.Alg_Bare 1.78 1.445 2.16
TREAT.Alg_NB 1.67 1.279 2.10
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 day.em <- emmeans(day.rstanarm3, ~TREAT, contr = list(TREAT = cmat), type = "link")$contrasts |>
gather_emmeans_draws() |>
mutate(Fit = exp(.value))
day.em |>
group_by(contrast) |>
mean_hdi()# A tibble: 4 × 10
contrast .value .value.lower .value.upper Fit Fit.lower Fit.upper .width
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 TREAT.Alg_B… 0.577 0.368 0.769 1.79 1.44 2.16 0.95
2 TREAT.Alg_NB 0.510 0.257 0.750 1.68 1.28 2.10 0.95
3 TREAT.Alg2_… 0.235 -0.0258 0.465 1.27 0.968 1.59 0.95
4 TREAT.NB_S 0.134 -0.184 0.466 1.16 0.797 1.55 0.95
# ℹ 2 more variables: .point <chr>, .interval <chr># A tibble: 4 × 2
contrast P
<chr> <dbl>
1 TREAT.Alg2_Alg1 0.966
2 TREAT.Alg_Bare 1
3 TREAT.Alg_NB 1
4 TREAT.NB_S 0.783## Probability of effect greater than 10%
day.em |>
group_by(contrast) |>
summarize(P = mean(Fit > 1.5))# A tibble: 4 × 2
contrast P
<chr> <dbl>
1 TREAT.Alg2_Alg1 0.0833
2 TREAT.Alg_Bare 0.946
3 TREAT.Alg_NB 0.783
4 TREAT.NB_S 0.0529# A tibble: 8 × 10
contrast .value .value.lower .value.upper Fit Fit.lower Fit.upper .width
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 TREAT.Alg_B… 0.579 0.449 0.713 1.78 1.55 2.02 0.8
2 TREAT.Alg_NB 0.512 0.349 0.678 1.67 1.40 1.95 0.8
3 TREAT.Alg2_… 0.238 0.0725 0.392 1.27 1.05 1.45 0.8
4 TREAT.NB_S 0.133 -0.0774 0.346 1.14 0.886 1.37 0.8
5 TREAT.Alg_B… 0.579 0.368 0.769 1.78 1.44 2.16 0.95
6 TREAT.Alg_NB 0.512 0.257 0.750 1.67 1.28 2.10 0.95
7 TREAT.Alg2_… 0.238 -0.0258 0.465 1.27 0.968 1.59 0.95
8 TREAT.NB_S 0.133 -0.184 0.466 1.14 0.797 1.55 0.95
# ℹ 2 more variables: .point <chr>, .interval <chr>ggplot(day.sum) +
geom_hline(yintercept = 1, linetype = "dashed") +
geom_pointrange(aes(x = contrast, y = Fit, ymin = Fit.lower, ymax = Fit.upper, size = factor(.width)),
show.legend = FALSE
) +
scale_size_manual(values = c(1, 0.5)) +
coord_flip()
g1 <- ggplot(day.sum) +
geom_hline(yintercept = 1) +
geom_pointrange(aes(x = contrast, y = Fit, ymin = Fit.lower, ymax = Fit.upper, size = factor(.width)), show.legend = FALSE) +
scale_size_manual(values = c(1, 0.5)) +
scale_y_continuous(trans = scales::log2_trans(), breaks = c(0.5, 1, 2, 4)) +
coord_flip() +
theme_classic()
g1
## Planned contrasts
cmat <- cbind(
"Alg2_Alg1" = c(-1, 1, 0, 0),
"NB_S" = c(0, 0, 1, -1),
"Alg_Bare" = c(0.5, 0.5, -0.5, -0.5),
"Alg_NB" = c(0.5, 0.5, -1, 0)
)
# On the link scale
day.brm3 |>
emmeans(~TREAT, type = "link") |>
contrast(method = list(TREAT = cmat))Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead. contrast estimate lower.HPD upper.HPD
TREAT.Alg2_Alg1 0.241 0.0168 0.501
TREAT.NB_S 0.123 -0.2197 0.428
TREAT.Alg_Bare 0.588 0.3668 0.788
TREAT.Alg_NB 0.528 0.2708 0.785
Point estimate displayed: median
Results are given on the log (not the response) scale.
HPD interval probability: 0.95 # On the response scale
day.brm3 |>
emmeans(~TREAT, type = "response") |>
contrast(method = list(TREAT = cmat))Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead. contrast ratio lower.HPD upper.HPD
TREAT.Alg2_Alg1 1.27 1.004 1.63
TREAT.NB_S 1.13 0.793 1.52
TREAT.Alg_Bare 1.80 1.408 2.16
TREAT.Alg_NB 1.70 1.278 2.15
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 day.em <- day.brm3 |>
emmeans(~TREAT, type = "link") |>
contrast(method = list(TREAT = cmat)) |>
gather_emmeans_draws() |>
mutate(Fit = exp(.value))Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead.# A tibble: 4 × 7
contrast Fit .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 TREAT.Alg_Bare 1.80 1.41 2.17 0.95 median hdi
2 TREAT.Alg_NB 1.70 1.30 2.18 0.95 median hdi
3 TREAT.Alg2_Alg1 1.27 0.982 1.61 0.95 median hdi
4 TREAT.NB_S 1.13 0.782 1.51 0.95 median hdi # A tibble: 4 × 2
contrast P
<chr> <dbl>
1 TREAT.Alg2_Alg1 0.974
2 TREAT.Alg_Bare 1
3 TREAT.Alg_NB 1
4 TREAT.NB_S 0.762# A tibble: 4 × 2
contrast P
<chr> <dbl>
1 TREAT.Alg2_Alg1 0.0912
2 TREAT.Alg_Bare 0.956
3 TREAT.Alg_NB 0.815
4 TREAT.NB_S 0.0408# A tibble: 8 × 7
contrast .value .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 TREAT.Alg_Bare 0.588 0.453 0.722 0.8 median hdci
2 TREAT.Alg_NB 0.528 0.367 0.703 0.8 median hdci
3 TREAT.Alg2_Alg1 0.241 0.0976 0.407 0.8 median hdci
4 TREAT.NB_S 0.123 -0.0695 0.364 0.8 median hdci
5 TREAT.Alg_Bare 0.588 0.367 0.788 0.95 median hdci
6 TREAT.Alg_NB 0.528 0.271 0.785 0.95 median hdci
7 TREAT.Alg2_Alg1 0.241 0.0168 0.501 0.95 median hdci
8 TREAT.NB_S 0.123 -0.220 0.428 0.95 median hdci g1 <- ggplot(day.sum) +
geom_vline(xintercept = 0, linetype = "dashed") +
geom_pointrange(
aes(
y = contrast, x = .value, xmin = .lower, xmax = .upper,
size = factor(.width)
),
show.legend = FALSE
) +
scale_size_manual(values = c(1, 0.5))
day.sum <- day.em |>
group_by(contrast) |>
median_hdci(Fit, .width = c(0.8, 0.95))
day.sum# A tibble: 8 × 7
contrast Fit .lower .upper .width .point .interval
<chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
1 TREAT.Alg_Bare 1.80 1.55 2.03 0.8 median hdci
2 TREAT.Alg_NB 1.70 1.41 1.98 0.8 median hdci
3 TREAT.Alg2_Alg1 1.27 1.07 1.47 0.8 median hdci
4 TREAT.NB_S 1.13 0.898 1.39 0.8 median hdci
5 TREAT.Alg_Bare 1.80 1.41 2.17 0.95 median hdci
6 TREAT.Alg_NB 1.70 1.30 2.18 0.95 median hdci
7 TREAT.Alg2_Alg1 1.27 0.982 1.61 0.95 median hdci
8 TREAT.NB_S 1.13 0.782 1.51 0.95 median hdci g1 <- ggplot(day.sum) +
geom_vline(xintercept = 1, linetype = "dashed") +
geom_pointrange(aes(y = contrast, x = Fit, xmin = .lower, xmax = .upper, size = factor(.width)), show.legend = FALSE) +
scale_size_manual(values = c(1, 0.5)) +
scale_x_continuous(trans = scales::log2_trans(), breaks = c(0.5, 0.8, 1, 1.2, 1.5, 2, 4)) +
theme_classic()
g1
g1a <-
day.em |>
ggplot() +
geom_vline(xintercept = 1, linetype = "dashed") +
# geom_vline(xintercept = 1.5, alpha=0.3, linetype = 'dashed') +
stat_slab(aes(
x = Fit, y = contrast,
fill = stat(ggdist::cut_cdf_qi(cdf,
.width = c(0.5, 0.8, 0.95),
labels = scales::percent_format()
))
), color = "black") +
scale_fill_brewer("Interval", direction = -1, na.translate = FALSE) +
scale_x_continuous("Effect",
trans = scales::log2_trans(),
breaks = c(0.5, 0.8, 1, 1.2, 1.5, 2, 4)
) +
scale_y_discrete("",
breaks = c("TREAT.NB_S", "TREAT.Alg2_Alg1", "TREAT.Alg_NB", "TREAT.Alg_Bare"),
labels = c("Nat. Bare vs Scapped", "Algae 1 vs 2", "Algae vs Nat. Bare", "Algae vs Bare")
) +
theme_classic()
g1 + g1a
library(ggridges)
day.em |>
ggplot() +
geom_density_ridges(aes(x = Fit, y = contrast), alpha = 0.4) +
geom_vline(xintercept = 1, linetype = "dashed")Picking joint bandwidth of 0.0366
day.em |>
ggplot() +
geom_density_ridges_gradient(
aes(
x = Fit,
y = contrast,
fill = stat(x)
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_c(option = "C")Picking joint bandwidth of 0.0364
day.em |>
ggplot() +
geom_density_ridges_gradient(
aes(
x = 100 * (Fit - 1),
y = contrast,
fill = stat(x)
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous("Percentage change") +
scale_fill_viridis_c(option = "C")Picking joint bandwidth of 3.66
## Or on a fractional scale
day.brm3 |>
gather_draws(`.*TREAT.*`, regex = TRUE) |>
ggplot() +
geom_density_ridges_gradient(
aes(
x = exp(.value),
y = .variable,
fill = stat(x)
),
alpha = 0.4, colour = "white",
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)
) +
geom_vline(xintercept = 1, linetype = "dashed") +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_c(option = "C")Picking joint bandwidth of 0.0388
11 Summary Figure
TREAT rate lower.HPD upper.HPD
ALG1 22.22805 18.28628 26.32820
ALG2 28.18155 23.51524 32.91237
NB 15.00118 11.92159 18.56181
S 13.18862 10.09135 16.34873
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 ## A quick version
g2 <- ggplot(newdata, aes(y = rate, x = TREAT)) +
geom_pointrange(aes(ymin = lower.HPD, ymax = upper.HPD)) +
theme_classic()
g2 + g1
Warning: Method 'parse_bf' is deprecated. Please use 'brmsterms' instead. TREAT rate lower.HPD upper.HPD
ALG1 22.31651 18.47330 26.71662
ALG2 28.43050 24.13011 33.27783
NB 14.90500 11.66270 18.50108
S 13.20072 10.11007 16.35482
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 ## A quick version
g2 <- ggplot(newdata, aes(y = rate, x = TREAT)) +
geom_pointrange(aes(ymin = lower.HPD, ymax = upper.HPD)) +
scale_y_continuous("Number of newly recruited barnacles") +
scale_x_discrete("",
breaks = c("ALG1", "ALG2", "NB", "S"),
labels = c("Algae 1", "Algae 2", "Nat. Bare", "Scraped")
) +
theme_classic()
g2 + g1
12 References
Quinn, G. P., and K. J. Keough. 2002. Experimental Design and Data Analysis for Biologists. London: Cambridge University Press.